For a real number y, let [y] denotes the greatest integer less than or equal to y. Let $f\left(x\right)=\frac{\tan \left(\pi \right[x-\pi \left]\right)}{1+[x{]}^2}$ Then

$f\left(x\right)$ is discontinuous at some $x$

$f\left(x\right)$ is continuous at all x, but the derivative $f^{\mathrm{\prime }}\left(x\right)$ does not exist for some $x$

$f^{\mathrm{\prime }}\left(x\right)$ exists for all $x$

$f^{\mathrm{\prime }}\left(x\right)$ exist for all x, but the derivative $f^{\mathrm{\prime }}\left(x_0\right)$ does not exist for some $x$